Your intuition is incorrect. They are both grammatical. And they are identical in meaning. The only difference between these two sentences is how many syllables they have.
The cluster to be, consisting of the infinitive complementizer to, plus the predicate adjective auxiliary infinitive be, is frequently deleted after the predicate seem (or appear) before a predicate adjective, like inseparable (meaning 'very close friends'). There is no specific rule saying when to perform this deletion; it's a matter of individual choice, like many other rules in English.
The reason why to be can be deleted here is that it has no meaning, and serves merely to mark the complement clause as an infinitive (required after seem) and the predicate of the complement clause as a predicate adjective (required before inseparable). So it's dispensable.
There are lots of syntactic rules (which means "processes", btw, and not rules for "Correctness" -- think of them as grammatical apps in your brain) in English that have the effect of shortening, moving, or deleting such frequently-occurring but semantically null chunks, and otherwise make speech faster. And supposedly easier.
Easier for the speaker, anyway. They don't always make things easier for the understander, or the learner, though. Frequently you have to put all that stuff back into the sentence to make it clear.
This rule (or app) is To-be-Deletion; a similar one for a different situation is Whiz Deletion.